2.1. Improved cohesive element constitution equation

The interphase layer is a chemical reaction-produced layer between the fiber and the matrix. Hence, the interphase layer is usually very thin, and cohesion is actually the internal force in the interphase layer from atom–molecule interactions. The cohesive element model, simplified in this case, can reflect the mechanical behaviors of the reaction product layer when the appropriate parameters are selected.

The cohesive element obeys the defined traction–separation law, as shown in Fig. 1, which assumes initially linear elastic behavior, followed by the initiation and evolution of damage (complete failure at u_f). The relation between strain and displacement u is shown below:

Fig. 1Full-screen View

Improved traction–separation law

where u is the relative displacement of the two contacting element faces attached to a cohesive element, and t is the constitutive thickness of the cohesive elements.

Usually, the traction–separation law is expressed in terms of displacement instead of strain. The initially linear elastic constitutive equation is shown below:

where $E$ and $K$ denote the modulus and stiffness, respectively, and σ is the stress.

Several initiation criteria and evolution laws are available in the ABAQUS modeling software [25]. Because the main failure mechanism during push-out testing of SiC composites is shear stress failure of the interface, the maximum shear stress failure criterion is used in this study:

where ${\tau }^{*}$ and ${\tau }_{\text{f}}$ denote the critical shear strength and sliding frictional stress, respectively, and $T_{\max }$ is the maximum shear stress. The cohesive element begins to fail when the shear stress reaches $T_{\max }$.

The damage evolution is based on the 1 − d reduction of the stiffness, where d is the damage variable. The stress ${\sigma }^f$ of the failure of cohesive elements is expressed as follows:

To ensure that the shear stress declines to the constant value $T_{\text{f}}$, equal to the sliding frictional stress, after the failure, the damage variable $d$ is given as

where u₀ and $u^{\text{'}}$ denote the initial and relative failure displacements, respectively, and c is a fitting constant. The damage variable as a function of the displacement is achieved through tabulation.

2.2. Modeling the fiber push-out test

The 2D axisymmetric finite element model shown Fig. 2 was proposed to simulate the single-fiber push-out test. In order to compare the experimental results with those reported by Zeng et al.[16], SiC/Ti-matrix composites were chosen to build the model. The model defines four zones: (1) the SiC fiber; (2) the cohesive zone; (3) the Ti matrix; (4) the average material surrounding the matrix, having effective composite properties calculated from the linear rule of mixtures. In this analysis, the SiC fibers and Ti matrix are regarded as perfectly isotropic elastic materials, while the average material is treated as a transverse isotropic elastic material. The slice height is 0.43 mm, SiC fiber radius is 71 µm (including an 18-µm-thick carbon core and a 3-µm-thick pyrolytic carbon protection layer), Ti matrix radius is 112 µm with a fiber volume fraction equal to 40%, and the outer radius of the composite slice is 1.75 mm. The radii of the indenter and the constraint boundary are 50 µm and 250 µm, respectively. The single-fiber push-out test model is established at 20 °C, 300 °C and 530 °C, using the material data given in Ref. [16]. The stress-free temperature is assumed to be 770 °C, which is the sample preparation temperature.

Fig. 2Full-screen View

(Color online) Finite element model of single-fiber push-out test for SiC/Ti composite

The finite element analysis was performed using ABAQUS. The elements of the fiber, matrix, and average material are CAX4 (four-node bilinear axisymmetric quadrilaterals); the elements of the cohesive zone are COHAX4 (four-node axisymmetric cohesive elements); and the elements of the indenter are RAX2 (two-node linear axisymmetric rigid links). Zero-thickness cohesive elements are embedded between the fiber and matrix elements. The chosen element size (including the cohesive element) is 6.32 µm in the z-direction. The fiber push-out analysis involves the following steps.

First, the sample is cooled from the stress-free temperature to the testing temperature; then, residual thermal stresses are introduced by the mismatch of thermal expansion coefficients between SiC and Ti. In this analysis, the cooling process is modeled by temperature loading from the initial temperature of 770 °C. The boundary conditions on Set 1 and Set 2 in this step are described as

where $U_z^{\text{f}}$, $U_z^{\text{m}}$, and $U_Z^{\text{ave}}$ represent the axial displacements of the fiber, matrix, and average material, respectively; $U_{\text{r}}$ represents the radial displacement; and h is the sample thickness. In addition, the free-end face of the specimen can be realized by removing the existing tying constraint. According to Ref. [22], an initial crack (length: 6.32 µm) is introduced at both ends of the specimen to realize the free end face of the specimen; this initial crack length is far smaller than the SiC fiber radius, and its addition does not affect the peak load.

The second step is the loading of a small displacement to establish stable contact between the rigid indenter and the fiber. The indenter is square instead of pyramidal to avoid stress concentration, and the contact is modeled as frictionless to improve the convergence; otherwise, the contact iteration solution could be disconcerted divergence by a sudden large load. The boundary condition on Set 3 in this step is described as

where $R^{\text{ave}}$ is the outer radius of the average material.

The third step is the modeling of the fiber push-out process. Fiber push-out is simulated with the displacement-controlled loading of a single fiber until this fiber is pushed out completely from the matrix. The cohesive elements start to fail when the shear stress reaches the maximum shear stress during the fiber push-out test; this means that the crack grows before the shear stress reaches a constant value, similar to the sliding frictional stress.

2.3. Verification of the improved cohesive element method

2.3.1. Damage evolution law and traction–separation relation

Since the failure mechanism is mainly shear failure, as mentioned earlier, only the shear damage evolution law is discussed here, although cases of both normal and shear damage are considered. The previously described damage evolution law is plotted in Fig. 3(a). The cohesive element does not fail before the initial failure displacement $u_0$, at which the shear stress is equal to the maximum shear stress, and the damage variable $d$ always remains equal to zero. The sudden increase in $d$ as it rapidly approaches 0.9 indicates the beginning of failure, meaning that debonding occurs, since d then increases very slowly from 0.9 to 1 (final failure).

Fig. 3Full-screen View

(Color online) Variation of damage variable (a) and traction (b) with separation distance at 20 °C

The relationship between the traction and the separation distance of the cohesive element is shown in Fig. 3(b). In the early stage of separation, the traction increases linearly with the separation distance. When the shear stress reaches the critical value, the interface falls off, and the traction drops to a constant value related to the temperature. Usually, the solution of this drastic change is difficult to converge. In this study, viscous regularization is introduced to improve the convergence.

2.3.2. Verification process

Fiber push-out tests were simulated using this improved cohesive element method. The validity of the method for simulating interfacial debonding and sliding friction should be confirmed, and the effects of some geometry and physical parameters on the mechanical behaviors of the composites were investigated. Figure 4(a) exhibits the experimental load–displacement curve of the push-out test of the SiC/Ti composites at 20 °C from Ref. [16]. The relation between load and displacement is nearly linear for loads below the maximum load P_max. P_i is approximately 17 N according to the displacement of the fiber protruding from the bottom of the slice after complete (instable) debonding. According to Ref. [11], the linear relation remains because of the very slow crack expansion. The interface debonds fully at P_max. Hereafter, the load decreases to the sliding frictional force from the existence of interfacial roughness.

Fig.4Full-screen View

(Color online) The experimental and computational Load-displacement curves at RT, which (a) is the push-out test [16] and (b) is the simulation under different indenter radii.

Under the critical shear strength ${\tau }^{*}$ of 500 MPa and the constant frictional stress $T_{\text{f}}$ of 84 MPa, the simulated load–displacement curves with various indenter radii at 20 °C are shown in Fig. 4(b). From the comparison of the experimental and computational curves, the simulated maximum load and frictional force show good agreement with the experimental values, and the P_i value of 15 N is similar to that in the experiment (17 N). As shown in Fig. 4(b), the simulated value is lower than the experimental value in Fig. 4(a), and the displacement increases with decreasing indenter radius R. The difference may be related to the shape selection of the indenter; in order to facilitate convergence, a square indenter is used in the simulation instead of the triangular indenter in the experiment.

Figure 5(a) describes the corresponding distribution of shear stress from the bottom to the top of the interface. It clearly demonstrates that the shear stress introduced by loading is negative; as a result, the bottom residual shear stress is larger and the top shear stress is smaller. The bottom of the interface starts to fail when the load reaches the initial failure load of 15 N; then, as the load increases, the cracks propagate from the bottom to the top. However, the propagation direction of the cracks relates to the height of the sample. According to Ananth et al. [11], a critical height of ~0.5–0.75 mm is observed in SiC/Ti composites; the crack propagates from the top to the bottom along the interface when the sample height exceeds this critical height.

Fig. 5Full-screen View

(Color online) Verification simulation: (a) Distribution of the shear stress, (b) Changes in energy with time, (c) Relationship between crack length and load

As mentioned earlier, the viscous coefficient is introduced to resolve the severe nonlinearity of the cohesive element after failure. Viscous regularization is discussed in detail in Ref. [25]. The variations in external work, total energy, internal energy, and viscous energy over time are shown in Fig. 5(b). Note that the time is calibrated by the analysis steps. The total energy increases from zero to 0.02 mJ only during the cooling stage as thermal energy converts to internal energy; the external work only increases during the fiber push-out. This means that the effect of the viscous coefficient introduced on the model can be ignored by comparing the internal energy and viscous energy.

Figure 5(c) shows the relationship between the crack length and the load. The crack does not propagate before the initial failure load, and crack propagation is based on whether sliding occurs. The load first increases quickly to ~15 N and then increases slowly until it reaches the maximum load (30.6 N). The crack becomes unstable when the crack tip reaches a distance equal to twice the fiber radius from the bottom of the interface, and the propagation speed is higher in this stage.

The experimental and simulated averaged values of the maximum load of the fiber/matrix interfacial area and the frictional shear stress are given in Table. 1. The simulated averaged maximum shear stresses and frictional shear stresses are in good agreement with the experimental results at 20 °C, 300°C, and 530°C. In addition, both the maximum and frictional shear stresses decrease with increasing test temperatures. All the compared results demonstrate that the improved cohesive element method can accurately simulate the interfacial debonding and sliding friction of SiC composites.

Table.1Full-screen View

Comparison of simulated averaged maximum and frictional shear stress with experimental values under different test temperatures

T (℃)	Averaged maximum shear stresses (MPa)			Averaged frictional shear stress (MPa)
	Exp.1[16]	Exp.2[16]	Simulation	Exp.1[16]	Exp.2[16]	Simulation
20	160	162	160	90	77	84
300	106	112	110	21	16	20
530	47	58	55	16	0	10

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3.1. Distribution of residual thermal stresses

The state of residual thermal stress is important in the fiber push-out test according to Zeng et al. [16] and Koss et al. [26], especially in metal matrix composites. The residual thermal stresses are introduced after cooling from the stress-free temperature to the testing temperature by the mismatched thermal expansion coefficients of SiC and Ti. Therefore, it is necessary to study the effects of the residual thermal stresses on the interfacial debonding and sliding friction.

The radial and axial residual thermal stresses at the center of the 430-µm-high slice in the radial direction at 20 °C are shown in Fig. 6(a). The radial stress σ_rr is negative, because compression occurs after cooling from high to low temperature. However, it becomes positive at the matrix/average material interface because their interaction is greater than the thermal effect. In the axial direction, the stress σ_zz is negative in the fiber but positive in the matrix, because the thermal expansion coefficient of the fiber is smaller than that of the matrix; thus, the fiber undergoes compressive stress while the matrix endures tensile stress. The distribution of residual shear stress at the interface is given in Fig. 6(b), indicating that the shear stress σ_rz presents an axially symmetric shape about the half-thickness of the sample, because of the axial displacement difference between the fiber and matrix at the interface. The maximum shear stress occurs at both ends of the specimen, which means that the crack is most likely to expand from both sides.

Fig. 6Full-screen View

(Color online) Distribution of residual thermal stresses in the middle of the slice in the radial direction (a) and residual shear stress from bottom to top along the interface (b)

3.2. Effect of temperature

Temperature is important in the process of fiber push-out, since the introduced residual thermal stresses greatly affect the interfacial behaviors of composites. For instance, both the interfacial debonding and sliding friction are dependent on the shear stress. Figure 7(a) indicates the distribution of shear stress from the bottom to top along the interface under different temperatures. At 20 °C, 300 °C, and 530 °C, the maximum residual shear stresses are 426 MPa, 225 MPa, and 105 MPa, respectively. The residual shear stress increases with decreasing temperature. The load–displacement curves under different temperatures are given in Fig. 7(b), showing that the frictional force and maximum load increases with decreasing temperature. It can be interpreted that the sliding frictional stress and maximum shear stress T_max increase with a decrease in temperature. Hence, the indenter force must be increased to overcome the interfacial adhesion and resistance, and the sliding frictional force increases after full debonding.

Fig. 7Full-screen View

(Color online) Effect of temperatures: (a) Distribution of shear stress at the interface, (b) Load–displacement curves under different temperatures

3.3. Effect of fracture strength

The critical shear strength reflects the adhesive strength between the fiber and matrix. In real applications, the cohesive strength cannot be too strong or too weak. Excessively strong cohesion easily causes fiber fracture before interfacial failure, and weak cohesion hinders the speed of load transfer from the matrix to the fiber. Figure 8 shows that the load required to cause complete interfacial failure increases with an increase in the critical shear strength, which means that the interfacial bond strength increases with the critical shear strength.

Fig. 8Full-screen View

(Color online) Load–displacement curves under different critical shear strengths

3.4. Effect of fiber volume fraction

The properties of a composite depend not only on its constituents but also on the volume fraction and distribution of fibers. In various processing methods, the fiber radius varies; the radius of chemical vapor deposition (CVD) SiC fibers is approximately 100 μm. Typically, hundreds of fibers are woven together during prefabrication. The weaving methods and fiber volume fractions are critical in determining the behaviors of composites. The effect of the fiber volume fraction on the initial failure load is shown in Table.2. The initial failure load increases with increasing fiber volume fraction, meaning that a composite material with a higher fiber volume fraction has a higher bearing capacity.

3.5. Effect of interphase layer

The interphase layer is a chemical reaction layer formed during the fiber impregnation process. The interfacial layer in SiC composites comprises TiC, especially when the SiC fiber has a pure C coating [27]. In this analysis, the interphase thickness is assumed as 3 µm, 6 µm, and 9 µm, respectively. Some data [28–30] suggest that the critical shear strength and sliding frictional stress increased significantly as the interfacial reaction proceeded and that the pyrolytic carbon layer on the SiC fibers was consumed. According to Table.3, the initial failure load increases with the interfacial thickness. This is consistent with the phenomenon reported in the literature. However, the effect of the interphase is limited because the overall composite behaviors also relate to the fiber volume fraction; thus, the thickness of the interphase should not be too large.